Matching with transfers: an economists toolbox Pierre-Andr Chiappori - - PowerPoint PPT Presentation

Matching with transfers: an economist’s toolbox

Pierre-André Chiappori

Columbia University

IIES, Stockholm, May 2013

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 1 / 76

Matching models: overview

Basic framework:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations;

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many ‘roommate’ matching (e.g. risk sharing)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many ‘roommate’ matching (e.g. risk sharing)

Goal: explain:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many ‘roommate’ matching (e.g. risk sharing)

Goal: explain:

Who is matched with whom?

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many ‘roommate’ matching (e.g. risk sharing)

Goal: explain:

Who is matched with whom? (in some models): how is the surplus allocated? ! therefore: endogeneize ‘power’ and intramatch allocations as functions of the ‘environment’

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many ‘roommate’ matching (e.g. risk sharing)

Goal: explain:

Who is matched with whom? (in some models): how is the surplus allocated? ! therefore: endogeneize ‘power’ and intramatch allocations as functions of the ‘environment’

Equilibrium concept: Stability

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many ‘roommate’ matching (e.g. risk sharing)

Goal: explain:

Who is matched with whom? (in some models): how is the surplus allocated? ! therefore: endogeneize ‘power’ and intramatch allocations as functions of the ‘environment’

Equilibrium concept: Stability

Robustness vis a vis bilateral deviations

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Matching models: overview

Basic framework:

2 heterogeneous populations; matching: one individual from each population Gain generated by such a match, match-speci…c

Generalizations:

many to one, many to many ‘roommate’ matching (e.g. risk sharing)

Goal: explain:

Who is matched with whom? (in some models): how is the surplus allocated? ! therefore: endogeneize ‘power’ and intramatch allocations as functions of the ‘environment’

Equilibrium concept: Stability

Robustness vis a vis bilateral deviations Interpretation: ‘divorce at will’

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 2 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...);

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...); impact on inequality, etc.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...); impact on inequality, etc.

... but also: how are the gain from marriage allocated?

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...); impact on inequality, etc.

... but also: how are the gain from marriage allocated? ... and: how does the market for marriage a¤ects individual and household behavior:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...); impact on inequality, etc.

... but also: how are the gain from marriage allocated? ... and: how does the market for marriage a¤ects individual and household behavior:

ex ante: human capital investment of future spouses (basic idea: HC improves marital prospects, in many directions)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...); impact on inequality, etc.

... but also: how are the gain from marriage allocated? ... and: how does the market for marriage a¤ects individual and household behavior:

ex ante: human capital investment of future spouses (basic idea: HC improves marital prospects, in many directions) ex post: human capital investment of existing couples (basic idea: expenditures may depend on the spouses’ respective ‘powers’ - cf collective model).

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...); impact on inequality, etc.

... but also: how are the gain from marriage allocated? ... and: how does the market for marriage a¤ects individual and household behavior:

ex ante: human capital investment of future spouses (basic idea: HC improves marital prospects, in many directions) ex post: human capital investment of existing couples (basic idea: expenditures may depend on the spouses’ respective ‘powers’ - cf collective model).

‘Tractable General Equilibrium’

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Possible interpretation: ‘marriage market’

Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom):

assortative matching (by education, income,...); impact on inequality, etc.

... but also: how are the gain from marriage allocated? ... and: how does the market for marriage a¤ects individual and household behavior:

ex ante: human capital investment of future spouses (basic idea: HC improves marital prospects, in many directions) ex post: human capital investment of existing couples (basic idea: expenditures may depend on the spouses’ respective ‘powers’ - cf collective model).

‘Tractable General Equilibrium’ Di¤erent models are better suited for some purposes than for others.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76

Issues related to matching: two examples

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Example 1: Assortative matching and inequality

Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76

Example 1: Assortative matching and inequality

Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76

Example 1: Assortative matching and inequality

Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching. Several questions; in particular:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76

Example 1: Assortative matching and inequality

Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching. Several questions; in particular:

Why did correlation change? Did ‘preferences for assortativeness’ change?

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76

Example 1: Assortative matching and inequality

Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching. Several questions; in particular:

Why did correlation change? Did ‘preferences for assortativeness’ change? How do we compare single-adult households and couples? What about intrahousehold inequality?

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76

Example 2: College premium and the demand for college education

Motivation: remarkable increase in female education, labor supply, incomes worldwide during the last decades.

Source: Becker-Hubbard-Murphy 2009

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 6 / 76

Example 2: College premium and the demand for college education

In the US:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 7 / 76

Example 2: College premium and the demand for college education

Questions: why such di¤erent responses by gender?

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 8 / 76

Example 2: College premium and the demand for college education

Questions: why such di¤erent responses by gender? impact on intrahousehold allocation?

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 8 / 76

Example 2: College premium and the demand for college education

Questions: why such di¤erent responses by gender? impact on intrahousehold allocation? impact on household behavior (expenditure, HC investment, etc.) ! especially relevant in developing countries!

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 8 / 76

Roadmap

1

Matching models: general presentation

2

The case of Transferable Utility (TU)

3

Extensions:

Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing

4

Econometric implementation

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 9 / 76

Roadmap

1

Matching models: general presentation

2

The case of Transferable Utility (TU)

3

Extensions:

Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing

4

Econometric implementation

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 10 / 76

Matching models: three main families

1

Matching under NTU (Gale-Shapley) Idea: no transfer possible between matched partners

2

Matching under TU (Becker-Shapley-Shubik)

Transfers possible without restrictions Technology: constant ‘exchange rate’ between utiles In particular: (strong) version of interpersonal comparison of utilities ! requires restrictions on preferences

3

Matching under Imperfectly TU (ITU)

Transfers possible But no restriction on preferences ! technology involves variable ‘exchange rate’

... plus ‘general’ approaches (’matching with contracts’, from Kelso-Crawford to Milgrom-Hat…eld-Kominers and friends) ... and links with: auction theory, general equilibrium.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 11 / 76

Uw Uh Pareto frontier: NTU

Uw Uh Pareto frontier: TU

Uw Uh Pareto frontier: ITU

Matching models: three main families

Similarities and di¤erences All aimed at understanding who is matched with whom Only the last 2 address how the surplus is divided Only the third allows for impact on the group’s aggregate behavior

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 12 / 76

Formal structure: Common components

Compact, separable metric spaces X, Y (‘women, men’) with …nite measures F and G. Note that the spaces may be multidimensional Spaces X, Y often ‘completed’ to allow for singles: ¯ X = X [ f∅g , ¯ Y = Y [ f∅g A matching de…nes of a measure h on X Y (or ¯ X ¯ Y ) such that the marginals of h are F and G The matching is pure if the support of the measure is included in the graph of some function φ Translation: matching is pure if y = φ (x) a.e. ! no ‘randomization’

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 13 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

NTU: two funtions u (x, y) , v (x, y)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

NTU: two funtions u (x, y) , v (x, y) TU: one function s (x, y) (intrapair allocation is endogenous)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

NTU: two funtions u (x, y) , v (x, y) TU: one function s (x, y) (intrapair allocation is endogenous) ITU: Pareto frontier u = F (x, y, v)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

NTU: two funtions u (x, y) , v (x, y) TU: one function s (x, y) (intrapair allocation is endogenous) ITU: Pareto frontier u = F (x, y, v)

De…ning the solution

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

NTU: two funtions u (x, y) , v (x, y) TU: one function s (x, y) (intrapair allocation is endogenous) ITU: Pareto frontier u = F (x, y, v)

De…ning the solution

NTU: only the measure h; stability as usual

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

NTU: two funtions u (x, y) , v (x, y) TU: one function s (x, y) (intrapair allocation is endogenous) ITU: Pareto frontier u = F (x, y, v)

De…ning the solution

NTU: only the measure h; stability as usual TU: measure h and two functions u (x) , v (y) such that u (x) + v (y) = s (x, y) for (x, y) 2 Supp (h) and stability u (x) + v (y) s (x, y) for all (x, y)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences

De…ning the problem: populations X, Y plus

NTU: two funtions u (x, y) , v (x, y) TU: one function s (x, y) (intrapair allocation is endogenous) ITU: Pareto frontier u = F (x, y, v)

De…ning the solution

NTU: only the measure h; stability as usual TU: measure h and two functions u (x) , v (y) such that u (x) + v (y) = s (x, y) for (x, y) 2 Supp (h) and stability u (x) + v (y) s (x, y) for all (x, y) ITU: measure h and two functions u (x) , v (y) such that u (x) = F (x, y, v (y)) for (x, y) 2 Supp (h) and stability u (x) F (x, y, v (y)) for all (x, y)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76

Formal structure: di¤erences (cont.)

Characterization:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

Stability equivalent to surplus maximization

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

Stability equivalent to surplus maximization therefore: existence easy to establish

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness

In a nutshell

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness

In a nutshell

NTU: intragroup allocation exogenously imposed; transfers are ruled

• ut by assumption

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness

In a nutshell

NTU: intragroup allocation exogenously imposed; transfers are ruled

• ut by assumption

TU and ITU: intragroup allocation endogenous; transfers are paramount and determined (or constrained) by equilibrium conditions

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Formal structure: di¤erences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c

Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness

In a nutshell

NTU: intragroup allocation exogenously imposed; transfers are ruled

• ut by assumption

TU and ITU: intragroup allocation endogenous; transfers are paramount and determined (or constrained) by equilibrium conditions TU: life much easier (GQL ! equivalent to surplus maximization) ... ... but price to pay: couple’s (aggregate) behavior does not depend on ‘powers’, therefore on equilibrium conditions

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76

Implications (crucial for empirical implementation)

NTU: stable matchings solve u(x) = max

z fU(x, z)jV (x, z) v(z)g

and v(y) = max

z fV (z, y)jU(z, y) u(z)g

for some pair of functions u and v.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 16 / 76

Implications (crucial for empirical implementation)

NTU: stable matchings solve u(x) = max

z fU(x, z)jV (x, z) v(z)g

and v(y) = max

z fV (z, y)jU(z, y) u(z)g

for some pair of functions u and v. TU: stable matchings solve u(x) = max

z fs(x, z) v(z)g and v(y) = max z fs(z, y) u(z)g

for some pair of functions u and v.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 16 / 76

Implications (crucial for empirical implementation)

NTU: stable matchings solve u(x) = max

z fU(x, z)jV (x, z) v(z)g

and v(y) = max

z fV (z, y)jU(z, y) u(z)g

for some pair of functions u and v. TU: stable matchings solve u(x) = max

z fs(x, z) v(z)g and v(y) = max z fs(z, y) u(z)g

for some pair of functions u and v. ITU: stable matchings solve u(x) = max

z fF(x, z, v (z))g and v(y) = max z fF 1(z, y, u (z))g

for some pair of functions u and v.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 16 / 76

Roadmap

1

Matching models: general presentation

2

The case of Transferable Utility (TU)

3

Extensions:

Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing

4

Econometric implementation

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 17 / 76

Transferable Utility (TU)

De…nition

A group satis…es TU if there exists monotone transformations of individual utilities such that the Pareto frontier is an hyperplane u (x) + v (y) = s (x, y) for all values of prices and income. ! Marriage market: assumption on preferences? Model: collective (public and private consumptions, e¢cient decisions) TU if ‘Generalized Quasi Linear (GQL, Bergstrom and Cornes 1981): ui (qi, Q) = Fi

• Ai
• q2

i , ..., qn i , Q

+ q1

i bi (Q)

• with bi (Q) = b (Q) for all i (much more general than QL)

Then standard model: x, y incomes and: s (x, y) = H (x + y) = max F 1

1

(u1) + F 1

2

(u2) under BC

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Basic result

If a matching is stable, the corresponding measure satis…es the surplus maximization problem, which is an optimal transportation problem (Monge-Kantorovitch): Find a measure h on X Y such that the marginals of h are F and G, and h solves max

h

Z

X Y s (x, y) dh (x, y)

Hence: linear programming

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 19 / 76

Basic result

If a matching is stable, the corresponding measure satis…es the surplus maximization problem, which is an optimal transportation problem (Monge-Kantorovitch): Find a measure h on X Y such that the marginals of h are F and G, and h solves max

h

Z

X Y s (x, y) dh (x, y)

Hence: linear programming Dual problem: dual functions u (x) , v (y) and solve min

u,v

Z

X u (x) dF (x) +

Z

Y v (y) dG (y)

under the constraint u (x) + v (y) s (x, y) for all (x, y) 2 X Y

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 19 / 76

Basic result

If a matching is stable, the corresponding measure satis…es the surplus maximization problem, which is an optimal transportation problem (Monge-Kantorovitch): Find a measure h on X Y such that the marginals of h are F and G, and h solves max

h

Z

X Y s (x, y) dh (x, y)

Hence: linear programming Dual problem: dual functions u (x) , v (y) and solve min

u,v

Z

X u (x) dF (x) +

Z

Y v (y) dG (y)

under the constraint u (x) + v (y) s (x, y) for all (x, y) 2 X Y In particular, the dual variables u and v describe an intrapair allocation compatible with a stable matching

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Links with hedonic models

Structure: three sets (‘buyers’ X, ‘sellers’ Y , ‘products’ Z) with measures µ, ν, σ. B Buyer x: quasi linear preferences U (x, z) P (z); seller y maximizes pro…t P (z) c (y, z) Equilibrium: price function P (z) that clear markets Technically: function P and measure α on the product set X Y Z such that (i) marginal of α on X (resp. Y ) coincides with µ (resp. ν) (ii) for all (x, y, z) in the support of α, U (x, z) P (z) = max

z 02K

• U
• x, z0 P
• z0

and P (z) c (y, z) = max

z 02K

• P
• z0 c
• y, z0

.

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Links with hedonic models

Chiappori, McCann and Nesheim (2010): canonical correspondance between QL hedonic models and matching models under TU. Speci…cally:

Consider a hedonic model and de…ne surplus: s(x, y) = max

z2Z (U(x, z) c(y, z))

Let η be the marginal of α over X Y , u (x) and v (y) by u (x) = max

z2K U (x, z) P (z) and v (y) = max z2K P (z) c (y, z)

Then (η, u, v) de…nes a stable matching Conversely, starting from a stable matching (η, u, v), u(x) + v(y) s (x, y) U (x, z) c (y, z) ) c (y, z) + v (y) U (x, z For any z, take P (z) such that inf

y 2J fc (y, z) + v (y)g P (z) sup x2I

fu (x, z) u (x)g then P (z) is an equilibrium price for the hedonic model.

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Supermodularity and assortative matching

Assume X, Y one-dimensional. Then s is supermodular if whenever x > x0 and y > y 0 then s (x, y) + s

• x0, y 0 > s
• x, y 0 + s
• x0, y
• Interpretation: single crossing (Spence - Mirrlees)

Consequence: matching is assortative Generalization (CMcCN ET 2010):

De…nition

A surplus function s : X Y ! [0, ∞[ is said to be Xtwisted if there is a set XL X0 of zero volume such that ∂xs(x0, y1) is disjoint from ∂xs(x0, y2) for all x0 2 X0 n XL and y1 6= y2 in Y . Then the stable matching is unique and pure

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Intracouple allocation under TU

Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares...

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Intracouple allocation under TU

Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations

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Intracouple allocation under TU

Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations However, basic result:

With a continuum of agents, intramatch allocation of welfare is pinned down by the equilibrium conditions

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Intracouple allocation under TU

Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations However, basic result:

With a continuum of agents, intramatch allocation of welfare is pinned down by the equilibrium conditions

Known from the outset, but ...

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 23 / 76

Intracouple allocation under TU

Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations However, basic result:

With a continuum of agents, intramatch allocation of welfare is pinned down by the equilibrium conditions

Known from the outset, but ... ... much easier than you would think

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Pinning down intracouple allocation under TU

Assume X, Y one dimensional and s supermodular. Then 3 steps Step 1: supermodularity implies assortative matching: x matched with y = ψ (x) if the number of women above x equals the number of men above ψ (x)

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Pinning down intracouple allocation under TU

Assume X, Y one dimensional and s supermodular. Then 3 steps Step 1: supermodularity implies assortative matching: x matched with y = ψ (x) if the number of women above x equals the number of men above ψ (x) Step 2: Stability implies u (x) = max

y

s (x, y) v (y) with the max being reached for y = ψ (x). Therefore u0 (x) = ∂s ∂x (x, ψ (x)) and v 0 (y) = ∂s ∂y (φ (y) , y) and u (x) = k +

Z x

∂s ∂x (t, ψ (t)) dt , v (y) = k0 +

Z y

∂s ∂y (φ (s) , s) ds ! Utilities de…ned up to two additive constants

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Pinning down intracouple allocation under TU

Step 3: pin down the constants

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Pinning down intracouple allocation under TU

Step 3: pin down the constants

Note that u (x) + v (ψ (x)) = s (x, ψ (x)) which pins down the sum k + k0

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Pinning down intracouple allocation under TU

Step 3: pin down the constants

Note that u (x) + v (ψ (x)) = s (x, ψ (x)) which pins down the sum k + k0 If one gender in excess supply (say women): the ‘last married’ woman indi¤erent between marriage and singlehood

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Pinning down intracouple allocation under TU

Step 3: pin down the constants

Note that u (x) + v (ψ (x)) = s (x, ψ (x)) which pins down the sum k + k0 If one gender in excess supply (say women): the ‘last married’ woman indi¤erent between marriage and singlehood Note: typically, discontinuity

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Pinning down intracouple allocation under TU

Step 3: pin down the constants

Note that u (x) + v (ψ (x)) = s (x, ψ (x)) which pins down the sum k + k0 If one gender in excess supply (say women): the ‘last married’ woman indi¤erent between marriage and singlehood Note: typically, discontinuity If equal number (knife-edge situation), indeterminate ... ... unless corner solutions

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Applications: applied theory

Various applications: Abortion and female empowerment (CO JPE 2006) Children and divorce (CW JoLE 2007) Male and female demand for higher education (CIW AER 2009) Dynamics: divorce and impact of divorce laws (CIW 10) Multidimensional matching:

general framework (Galichon Salanié 2011) income/education and physical attractiveness (COQ 2011) income and smoking habits (COQ 2012) income and ‘reproductive capital’ (Low 2012)

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Roadmap

1

Matching models: general presentation

2

The case of Transferable Utility (TU)

3

Extensions

Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing

4

Econometric implementation

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Extensions: roadmap

1

Pre-investment

2

Multidimensional matching

Theory Practical Implementation

3

ITU

General presentation A speci…c model

4

Risk sharing

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 28 / 76

Extensions: roadmap

1

Pre-investment

2

Multidimensional matching

Theory Practical Implementation

3

ITU

General presentation A speci…c model

4

Risk sharing

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Pre-investment

Two stage game:

1

Agents independently (non cooperatively) invest in characteristics (say in HC)

2

Agents match on these characteristics Model solved backwards: For given distributions of characteristics, matching equilibrium pins down the allocation of the surplus This allocation de…nes the return from the …rst period investment ’Rational expectations’: the distribution of characteristics expected by the agents when investing is realized by their investment

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Is pre-investment e¢cient?

Two opposite arguments:

1

(‘Free rider’): My investment will increase the joint surplus, some of which goes to my (future) partner ! under investment

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Is pre-investment e¢cient?

Two opposite arguments:

1

(‘Free rider’): My investment will increase the joint surplus, some of which goes to my (future) partner ! under investment

2

(’Rat race’): I am competing again other potential spouses, I have to be better ! over investment

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 31 / 76

Is pre-investment e¢cient?

Two opposite arguments:

1

(‘Free rider’): My investment will increase the joint surplus, some of which goes to my (future) partner ! under investment

2

(’Rat race’): I am competing again other potential spouses, I have to be better ! over investment

3

In fact: the investment is e¢cient Why? u0 (x) = ∂s ∂x (x, ψ (x))

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 31 / 76

Is pre-investment e¢cient?

Two opposite arguments:

1

(‘Free rider’): My investment will increase the joint surplus, some of which goes to my (future) partner ! under investment

2

(’Rat race’): I am competing again other potential spouses, I have to be better ! over investment

3

In fact: the investment is e¢cient Why? u0 (x) = ∂s ∂x (x, ψ (x))

4

Application: gender unbalance: who invests more? (ACM)

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Extensions: roadmap

1

Pre-investment

2

Multidimensional matching

Theory Practical Implementation

3

ITU

General presentation A speci…c model

4

Risk sharing

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Two-dimensional example: X R2, Y R2 Surplus S (x1, x2, y1, y2) Particular case (‘index’): S (x1, x2, y1, y2) = S (A (x1, x2) , B (y1, y2)) Two questions:

1

Who marries whom?

2

How is the surplus shared?

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Who marries whom?

Two possible approaches:

1

‘Guess’ what the matching patterns will look like; then:

Compute the thresholds Compute the individual utilities (see below) Check the stability conditions

2

Use surplus maximization

Always possible Typically: optimal control Very useful for simulations, etc.

Common caveat: matching may not be ‘pure’

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Purity

Idea: generalize the one-dimensional ‘supermodularity ) assortativeness’ result Generalization of supermodularity (CMcCN ET 2010):

De…nition

A surplus function S : X Y ! [0, ∞[ is said to be Xtwisted if there is a set XL X0 of zero volume such that ∂xS(x0, y1) is disjoint from ∂xS(x0, y2) for all x0 2 X0 n XL and y1 6= y2 in Y . Then the stable matching is unique and pure

De…nition

The matching is pure if the measure h is born by the graph of a function: for almost all x there exists exactly one y such that x matched with y. If not: ‘randomization’: an open set of (say) women are indi¤erent between several men

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Who marries whom? (cont.)

Assume the condition is satis…ed: (y1, y2) = φ (x1, x2). Then surplus maximization: max

φ

Z

X S (x1, x2, φ (x1, x2)) dF (x1, x2)

with a constraint: The push-forward of F through φ coincides with G where the push-forward φ#F of F through φ de…ned by φ#F (B) = F

• φ1 (B)
• for any Borel B X

! Optimal control

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Sharing the surplus

As previously, 3 steps Step 1: (x1, x2) matched with (y1, y2) = ψ (x1, x2)

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Sharing the surplus

As previously, 3 steps Step 1: (x1, x2) matched with (y1, y2) = ψ (x1, x2) Step 2: Stability implies u (x1, x2) = max

y1,y2 S (x1, x2, y1, y2) v (y1, y2)

with the max being reached for y = ψ (x). Then 1st OC ∂u ∂xi = ∂S ∂xi (x1, x2, ψ (x1, x2)) The PDE must be compatible: ∂ ∂x2 ∂S ∂x1 (x1, x2, ψ (x1, x2))

• =

∂ ∂x1 ∂S ∂x2 (x1, x2, ψ (x1, x2))

• If so, utilities de…ned up to one additive constant (and same for men)

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Sharing the surplus

As previously, 3 steps Step 1: (x1, x2) matched with (y1, y2) = ψ (x1, x2) Step 2: Stability implies u (x1, x2) = max

y1,y2 S (x1, x2, y1, y2) v (y1, y2)

with the max being reached for y = ψ (x). Then 1st OC ∂u ∂xi = ∂S ∂xi (x1, x2, ψ (x1, x2)) The PDE must be compatible: ∂ ∂x2 ∂S ∂x1 (x1, x2, ψ (x1, x2))

• =

∂ ∂x1 ∂S ∂x2 (x1, x2, ψ (x1, x2))

• If so, utilities de…ned up to one additive constant (and same for men)

Step 3: pin down the constants

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Example: smoking (COQ 2011)

Setting: Two populations (men and women) of equal size, normalized to one. Socio-economic status: continuous variables x and y, uniformly distributed over [0, 1] Smoking: dichotomic, independent of status; kM and kW proportions

• f smokers

Surplus: Σ = s (x, y) if both spouses do not smoke Σ = λs (x, y) otherwise, λ < 1 In practice s (x, y) = (x + y)2 /2

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Example: smoking (COQ 2011)

Basic remark: The ‘twisted’ condition does not hold. Woman, index x0, non smoker: ∂xΣ = (x0 + y1) if she marries a non smoker with index y1 ∂xΣ = λ (x0 + y2) if she marries a smoker with index y2. For any y2 2 h

(1λ)x0 λ

, 1 i , if y1 = λy2 (1 λ) x0, then the couples (x0, y1) and (x0, y2) violate the twisted buyer condition; works for an open set of values x0 - namely x0 2

• 0,

λ 1λ

• .

Consequence: The stable matching may not be pure.

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The model

Particular case: if kM = kW then: All smoking women marry smoking men, and conversely All non smoking women marry non smoking men, and conversely In words: Even if λ very close to 1, fully discriminated submarkets But: in practice, kM > kW

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Method 1: surplus maximization

Four categories: fNW , SW , NM, SMg For each, let PA (t) denote the proba that an individual with income t marries a smoker ‘Push-forward’ condition:

assortative matching on income within each cell 8x 2 NW , let φNW (x) denote the income of the non smoking

• husband. Then

Z 1

x (1 PNW (t)) dFNW (t) =

Z 1

φNW (x) (1 PNM (t)) dGNM (t)

which pins down φNW (x); etc.

Finally, total surplus: Σ =

Z 1

0 (1 PNW (t)) S (t, φNW (t)) dFNW (t)

+

Z 1

0 PNW (t) λS (t, φNW (t)) dFNW (t) + ...

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Method 2: ‘Guessing’ the form of the result

Here:

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Method 2: ‘Guessing’ the form of the result

Then: Compute the utilities in each case Compute the thresholds (indi¤erence conditions) Check stability (can be done directly using the inequality conditions)

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Particular case

Assume that S (x1, x2, y1, y2) = Σ (A (x1, x2) , B (y1, y2)) Then:

• ne dimensional matching

but: depends on an index that is not known Basic intuition: two agents with the same index are equivalent for all potential partners; therefore they should have the same distribution of matches (i.e.: the measure h only depends on A and B). Consequence: the MRS ∂A/∂x1

∂A/∂x2 can be identi…ed

utility only depends on the index

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Extensions: roadmap

1

Pre-investment

2

Multidimensional matching

Theory Practical Implementation

3

ITU

General presentation A speci…c model

4

Risk sharing

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Imperfectly transferable utility: theory

Motivation Limitation of TU models: all Pareto optimums correspond to the same aggregate behavior Therefore, redistributing power between men and women cannot impact the structure of expenditures ‘Collective’ literature: important phenomenon

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Imperfectly transferable utilities

General case: Transfers possible... ... but the ‘exchange rate’ is not constant. In practice: u (x) = P (x, y, v (y)) with P decreasing in v, usually increasing in x and y. Stability: u (x) P (x, y, v (y)) 8x 2 X, y 2 Y But: no longer equivalent to a maximization (‘total surplus ’ not de…ned).

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Imperfectly transferable utility: theory

Stability u (x) max

y

P (x, y, v (y)) and equality if marriage probability positive. Hence: u (x) = max

y

P (x, y, v (y)) 1st O C: ∂P ∂y (x, y, v (y)) + v 0 (y) ∂P ∂v (x, y, v (y)) = 0 satis…ed for x = φ (y)

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Imperfectly transferable utility: theory

Stability u (x) max

y

P (x, y, v (y)) and equality if marriage probability positive. Hence: u (x) = max

y

P (x, y, v (y)) 1st O C: ∂P ∂y (x, y, v (y)) + v 0 (y) ∂P ∂v (x, y, v (y)) = 0 satis…ed for x = φ (y) Knowing φ, if ∂P/∂y > 0, v de…ned up to a constant by: v 0 (y) =

∂P ∂y (φ (y) , y, v (y)) ∂P ∂v (φ (y) , y, v (y)) > 0

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Imperfectly transferable utility: theory

Assortativity 1st OC: H (y, φ (y)) = 0 8y where H (y, x) = ∂P ∂y (x, y, v (y)) + v 0 (y) ∂P ∂v (x, y, v (y)) . therefore ∂H ∂y + ∂H ∂x φ0 (y) = 0 8y,

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Imperfectly transferable utility: theory

Assortativity 1st OC: H (y, φ (y)) = 0 8y where H (y, x) = ∂P ∂y (x, y, v (y)) + v 0 (y) ∂P ∂v (x, y, v (y)) . therefore ∂H ∂y + ∂H ∂x φ0 (y) = 0 8y, 2nd OC: ∂H ∂y 0 , ∂H ∂x φ0 (y) 0.

• r:

∂2P ∂x∂y (φ (y) , y, v (y)) + v 0 (y) ∂2P ∂x∂v (φ (y) , y, v (y))

• φ0 (y) 0 8y.

(1)

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Imperfectly transferable utility: theory

Assortative: φ0 (y) 0 therefore ∂2P ∂x∂y (φ (y) , y, v (y)) + v 0 (y) ∂2P ∂x∂v (φ (y) , y, v (y)) 0 8y. (2)

• r:

∂2P ∂x∂y (φ (y) , y, v (y))

∂P ∂y (φ (y) , y, v (y)) ∂P ∂v (φ (y) , y, v (y))

∂2P ∂x∂v (φ (y) , y, v (y)) 0 8y. (3) TU case: P (x, y, v (y)) = s (x, y) v (y), hence

∂2P ∂x∂v = 0 and condition

∂2P ∂x∂y = ∂2s ∂x∂y 0

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Imperfectly transferable utility: a speci…c model

Goal: capture two notions: spouses value the public good di¤erently (endogenous) changes in ‘powers’ a¤ect the structure of expenditures Model: Continuum of men and women; x, y incomes 1 public good, 1 private good Translation of distributions: matching functions (assuming assortativeness) are φ (y) = (y + β) /α and ψ (x) = αx β. Male preferences: um = cmQ Female preferences: uf (cf ) = ∞ if cf < ¯ c = cf + Q if cf ¯ c In particular, e¢ciency implies cf = ¯ c

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Pareto frontier

Note: uf ((x + y) + ¯ c) /2 The Pareto frontier: um = P ((x + y) , uf ) = (uf ¯ c) ((x + y) uf ) ,

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 1 2 3 4

u v

Figure: Frontière de Pareto

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Assortativeness

Here ∂P (x + y, v) ∂ (x + y) = v ¯ c, ∂P (x + y, v) ∂v = (2v (¯ c + (x + y))) therefore ∂2P (x + y, v) ∂ (x + y)2 = 0 and ∂2P (x + y, v) ∂ (x + y) ∂v = 1

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Allocation

We have that v 0 (y) =

∂P ∂y (φ (y) + y, v (y)) ∂P ∂v (φ (y) , y, v (y)) =

αv (y) α¯ c 2αv (y) (α + 1) y (α¯ c + β). Solution: let ω be the inverse of v, the equation becomes: ω0 (v) + (α + 1) αv α¯ c ω (v) = 2αv (α¯ c + β) αv α¯ c , Solution: ω (v) = K (v ¯ c) α+1

α +

2α 2α + 1v β + ¯ cα + 2αβ (α + 1) (2α + 1),

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Utilities and consumptions

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 1 2 3 4 5 6 7 8 9

x Utilities Figure: Utilities and consumptions

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Comparative statics

Start from λ = .8, and two scenarios:

1

Increase all female incomes by 25%, male unchanged

2

Increase all male incomes by 20%, female unchanged Note that: ‘Who marries whom’ unchanged

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Comparative statics

Start from λ = .8, and two scenarios:

1

Increase all female incomes by 25%, male unchanged

2

Increase all male incomes by 20%, female unchanged Note that: ‘Who marries whom’ unchanged Couples’ total income unchanged

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Comparative statics

Start from λ = .8, and two scenarios:

1

Increase all female incomes by 25%, male unchanged

2

Increase all male incomes by 20%, female unchanged Note that: ‘Who marries whom’ unchanged Couples’ total income unchanged In particular, under TU, no impact on expenditures

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 56 / 76

Comparative statics

Start from λ = .8, and two scenarios:

1

Increase all female incomes by 25%, male unchanged

2

Increase all male incomes by 20%, female unchanged Note that: ‘Who marries whom’ unchanged Couples’ total income unchanged In particular, under TU, no impact on expenditures But (presumably) here changes in powers

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‐1 1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 16 18 20 Q_1 c_m_1 Q_3 c_m_3

200 400 600 800 1000 1200 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 45 50 mu_1 mu_3 U_female_1 U_female_3 U_male_1 U_male_3

Extensions: roadmap

1

Pre-investment

2

Multidimensional matching

Theory Practical Implementation

3

ITU

General presentation A speci…c model

4

Risk sharing

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Risk sharing

Two brief points:

1

Matching under TU may apply to risk sharing ...

2

... but you still want to allow for ITU

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Risk sharing: S-W 2000

Utilities: CRRA Um = c1η

m

1 η , Uf = c1η

f

1 η

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Risk sharing: S-W 2000

Utilities: CRRA Um = c1η

m

1 η , Uf = c1η

f

1 η Expected utility E (Um) =

Z

c1η

m

1 η dF (cm) , E (Uf ) =

Z

c1η

f

1 η dG (cf )

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Risk sharing: S-W 2000

Utilities: CRRA Um = c1η

m

1 η , Uf = c1η

f

1 η Expected utility E (Um) =

Z

c1η

m

1 η dF (cm) , E (Uf ) =

Z

c1η

f

1 η dG (cf ) E¢cient risk sharing: cm = ky, cf = (1 k) y therefore E (Um) = k1η 1 η

Z

y1ηdF (y) , E (Uf ) = (1 k)1η 1 η

Z

y1ηdF (y) and we have TU: [(1 η) E (Um)]

1 1η + [(1 η) E (Uf )] 1 1η =

Z y1ηdF (y)

• 1

1η

= S

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Roadmap

1

Matching models: general presentation

2

The case of Transferable Utility (TU)

3

Extensions:

multidimensional matching Imperfectly Transferable Utility

4

Econometric implementation

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 61 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one Two solutions to reconcile:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one Two solutions to reconcile:

Introduce frictions ! search models (labor; marriage: Robin-Jacquemet, Gousse,...)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one Two solutions to reconcile:

Introduce frictions ! search models (labor; marriage: Robin-Jacquemet, Gousse,...) Keep the frictionless framework but introduce other, unobservable dimension (‘unobservable heterogeneity’)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one Two solutions to reconcile:

Introduce frictions ! search models (labor; marriage: Robin-Jacquemet, Gousse,...) Keep the frictionless framework but introduce other, unobservable dimension (‘unobservable heterogeneity’)

Here: explore the second path

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one Two solutions to reconcile:

Introduce frictions ! search models (labor; marriage: Robin-Jacquemet, Gousse,...) Keep the frictionless framework but introduce other, unobservable dimension (‘unobservable heterogeneity’)

Here: explore the second path Today: what can we identify from matching data only? ! note that more information may be available

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one Two solutions to reconcile:

Introduce frictions ! search models (labor; marriage: Robin-Jacquemet, Gousse,...) Keep the frictionless framework but introduce other, unobservable dimension (‘unobservable heterogeneity’)

Here: explore the second path Today: what can we identify from matching data only? ! note that more information may be available

about transfers (! hedonic models)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 62 / 76

Econometric implementation

Basic empirical issue: Theory predicts ’mechanical’ assortative matching In practice: correlation, but not equal to one Two solutions to reconcile:

Introduce frictions ! search models (labor; marriage: Robin-Jacquemet, Gousse,...) Keep the frictionless framework but introduce other, unobservable dimension (‘unobservable heterogeneity’)

Here: explore the second path Today: what can we identify from matching data only? ! note that more information may be available

about transfers (! hedonic models) about the outcome and/or the sharing (! collective model)

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Econometric implementation

Assume population divided into large ‘classes’ (e.g. by education)

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Econometric implementation

Assume population divided into large ‘classes’ (e.g. by education) Basic insight: unobserved characteristics (heterogeneity) ! Gain gIJ

ij generated by the match i 2 I, j 2 J:

gIJ

ij = Z IJ + εIJ ij

where I = 0, J = 0 for singles, and εIJ

ij random shock with mean zero.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 63 / 76

Econometric implementation

Assume population divided into large ‘classes’ (e.g. by education) Basic insight: unobserved characteristics (heterogeneity) ! Gain gIJ

ij generated by the match i 2 I, j 2 J:

gIJ

ij = Z IJ + εIJ ij

where I = 0, J = 0 for singles, and εIJ

ij random shock with mean zero.

Therefore: dual variables (ui, vj) also random (endogenous distribution). Problem: nothing is known (in general) about the dual distribution.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 63 / 76

Econometric implementation

Assume population divided into large ‘classes’ (e.g. by education) Basic insight: unobserved characteristics (heterogeneity) ! Gain gIJ

ij generated by the match i 2 I, j 2 J:

gIJ

ij = Z IJ + εIJ ij

where I = 0, J = 0 for singles, and εIJ

ij random shock with mean zero.

Therefore: dual variables (ui, vj) also random (endogenous distribution). Problem: nothing is known (in general) about the dual distribution. Stability: constrained by the inequalities ui + vj gIJ

ij

for any (i, j) ! large number (one inequality per potential couple) ... of which a few are in fact equalities

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Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles Marriage market (‘marital college premium’) ! several components:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles Marriage market (‘marital college premium’) ! several components:

Marriage probability

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles Marriage market (‘marital college premium’) ! several components:

Marriage probability Spouse’s (distribution of) education

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles Marriage market (‘marital college premium’) ! several components:

Marriage probability Spouse’s (distribution of) education Surplus generated

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles Marriage market (‘marital college premium’) ! several components:

Marriage probability Spouse’s (distribution of) education Surplus generated Distribution of the surplus

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles Marriage market (‘marital college premium’) ! several components:

Marriage probability Spouse’s (distribution of) education Surplus generated Distribution of the surplus

Simple framework: Total college premium as the sum of these two components; CIW’s story: huge discrepancies between genders regarding MCP

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Basic issue: why did female demand for education exceed male?

Possible explanation (CIW 2009): impact of education is twofold: Labor market (‘college premium’): higher wages, lower unemployment, better career prospects,... ! no huge di¤erence between men and women (if anything against women) and between couples and singles Marriage market (‘marital college premium’) ! several components:

Marriage probability Spouse’s (distribution of) education Surplus generated Distribution of the surplus

Simple framework: Total college premium as the sum of these two components; CIW’s story: huge discrepancies between genders regarding MCP Matching models adequate to distinguish, since they take singlehood as a benchmark

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 64 / 76

Theoretical model (CIW 2009)

Two-dimensional heterogeneity: willingness to marry and cost of acquiring education Two stage model:

Stage 1: choose education level and entry on the marriage market Stage 2: matching game

Resolution: backwards

solve matching for given population ! dual variables: expected utility for each education level then models decision to acquire education/enter the marriage market Note: …xed point

Problem 1: how to empirically estimate the second stage? Problem 2 (more ambitious): estimate the two stage model (ongoing work with M. Costa and C. Meghir)

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Econometric implementation: a structural model (CSW 2010)

Finite number of classes (here education): f1, ..., Mg for women, f1, ..., Ng for men

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Econometric implementation: a structural model (CSW 2010)

Finite number of classes (here education): f1, ..., Mg for women, f1, ..., Ng for men Unobserved heterogeneity (random preferences) ! for i 2 I, utility is ui (qi, Q) + αJ

i where αi =

• α1

i , ..., αN i

• P.A. Chiappori (Columbia University)

Matching with Transfers IIES, Stockholm, May 2013 66 / 76

Econometric implementation: a structural model (CSW 2010)

Finite number of classes (here education): f1, ..., Mg for women, f1, ..., Ng for men Unobserved heterogeneity (random preferences) ! for i 2 I, utility is ui (qi, Q) + αJ

i where αi =

• α1

i , ..., αN i

• Interpretation:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 66 / 76

Econometric implementation: a structural model (CSW 2010)

Finite number of classes (here education): f1, ..., Mg for women, f1, ..., Ng for men Unobserved heterogeneity (random preferences) ! for i 2 I, utility is ui (qi, Q) + αJ

i where αi =

• α1

i , ..., αN i

• Interpretation:

each female i 2 I draws a vector αi =

• α1

i , ..., αN i

• f

preferences/attractiveness (for levels of husband’s education)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 66 / 76

Econometric implementation: a structural model (CSW 2010)

Finite number of classes (here education): f1, ..., Mg for women, f1, ..., Ng for men Unobserved heterogeneity (random preferences) ! for i 2 I, utility is ui (qi, Q) + αJ

i where αi =

• α1

i , ..., αN i

• Interpretation:

each female i 2 I draws a vector αi =

• α1

i , ..., αN i

• f

preferences/attractiveness (for levels of husband’s education) same for men: βj =

• β1

j , ..., βM j

• P.A. Chiappori (Columbia University)

Matching with Transfers IIES, Stockholm, May 2013 66 / 76

Econometric implementation: a structural model (CSW 2010)

Finite number of classes (here education): f1, ..., Mg for women, f1, ..., Ng for men Unobserved heterogeneity (random preferences) ! for i 2 I, utility is ui (qi, Q) + αJ

i where αi =

• α1

i , ..., αN i

• Interpretation:

each female i 2 I draws a vector αi =

• α1

i , ..., αN i

• f

preferences/attractiveness (for levels of husband’s education) same for men: βj =

• β1

j , ..., βM j

• The sum αi + βj contributes to the surplus

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 66 / 76

Econometric implementation: a structural model (CSW 2010)

Finite number of classes (here education): f1, ..., Mg for women, f1, ..., Ng for men Unobserved heterogeneity (random preferences) ! for i 2 I, utility is ui (qi, Q) + αJ

i where αi =

• α1

i , ..., αN i

• Interpretation:

each female i 2 I draws a vector αi =

• α1

i , ..., αN i

• f

preferences/attractiveness (for levels of husband’s education) same for men: βj =

• β1

j , ..., βM j

• The sum αi + βj contributes to the surplus

Note that E [αi j i 2 I] = aI 6= 0 in general: αJ

i = aJ i + ˜

αJ

i with E

• ˜

αJ

i

• = 0

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Econometric implementation: a structural model (CSW 2010)

‘Second stage’: match after education has been chosen but before incomes are known ! economic surplus if i 2 I marries j 2 J: SIJ = E [s (x, y) j i 2 I, j 2 J] where s (x, y) de…ned as before

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Econometric implementation: a structural model (CSW 2010)

‘Second stage’: match after education has been chosen but before incomes are known ! economic surplus if i 2 I marries j 2 J: SIJ = E [s (x, y) j i 2 I, j 2 J] where s (x, y) de…ned as before Total surplus: sij = SIJ + αJ

i + βI j

= Z IJ + ˜ αJ

i + ˜

β

I j

where I = 0, J = 0 for singles, α0

i = β0 j = 0 by normalization,

Z IJ = SIJ + aJ

i + bI j and E

˜ αJ

i

= E h ˜ β

I j

i = 0

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Econometric implementation

The model sati…es a crucial identifying assumption (Choo-Siow 2006) Assumption S (separability): the idiosyncratic component εij is additively separable: εIJ

ij = αIJ i + βIJ j

(S) where E

• αIJ

i

= E h βIJ

j

i = 0.

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Econometric implementation

The model sati…es a crucial identifying assumption (Choo-Siow 2006) Assumption S (separability): the idiosyncratic component εij is additively separable: εIJ

ij = αIJ i + βIJ j

(S) where E

• αIJ

i

= E h βIJ

j

i = 0. Then:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 68 / 76

Econometric implementation

The model sati…es a crucial identifying assumption (Choo-Siow 2006) Assumption S (separability): the idiosyncratic component εij is additively separable: εIJ

ij = αIJ i + βIJ j

(S) where E

• αIJ

i

= E h βIJ

j

i = 0. Then:

Theorem

Under S, there exists UIJ and V IJ such that UIJ + V IJ = Z IJ and for any match (i 2 I, j 2 J) ui = UIJ + αIJ

i

vj = V IJ + βIJ

j

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Econometric implementation

Theorem

A NSC for i 2 I being matched with a spouse in J is: UIJ + αIJ

i

• UI0 + αI0

i

UIJ + αIJ

i

• UIK + αIK

i

for all K

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Econometric implementation

Theorem

A NSC for i 2 I being matched with a spouse in J is: UIJ + αIJ

i

• UI0 + αI0

i

UIJ + αIJ

i

• UIK + αIK

i

for all K In practice:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 69 / 76

Econometric implementation

Theorem

A NSC for i 2 I being matched with a spouse in J is: UIJ + αIJ

i

• UI0 + αI0

i

UIJ + αIJ

i

• UIK + αIK

i

for all K In practice:

take singlehood as a benchmark (interpretation!)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 69 / 76

Econometric implementation

Theorem

A NSC for i 2 I being matched with a spouse in J is: UIJ + αIJ

i

• UI0 + αI0

i

UIJ + αIJ

i

• UIK + αIK

i

for all K In practice:

take singlehood as a benchmark (interpretation!) assume the αIJ

i

are extreme value distributed

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Econometric implementation

Theorem

A NSC for i 2 I being matched with a spouse in J is: UIJ + αIJ

i

• UI0 + αI0

i

UIJ + αIJ

i

• UIK + αIK

i

for all K In practice:

take singlehood as a benchmark (interpretation!) assume the αIJ

i

are extreme value distributed then logit and expected utility: ¯ uI = E

• max

J

• UIJ + αIJ

i

• = ln

∑

J

exp UIJ + 1 ! = ln

• aI0

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Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

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Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education

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Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

Non testable (no OIR)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

Non testable (no OIR) Relies on strong assumptions

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

Non testable (no OIR) Relies on strong assumptions In particular, homoskedasticity hard to justify.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

Non testable (no OIR) Relies on strong assumptions In particular, homoskedasticity hard to justify.

Possible solution:

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

Non testable (no OIR) Relies on strong assumptions In particular, homoskedasticity hard to justify.

Possible solution:

consider several ‘markets’ (here cohorts), with di¤erent marginals (composition by education classes)

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

Non testable (no OIR) Relies on strong assumptions In particular, homoskedasticity hard to justify.

Possible solution:

consider several ‘markets’ (here cohorts), with di¤erent marginals (composition by education classes) assume ‘some’ invariance across cohorts

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 70 / 76

Identi…cation: static model (Choo Siow 2006)

The model is exactly identi…ed:

Data: marriage matrix by classes of education One-to-one correspondance between that matrix and the Z matrix Although: could add covariates

Once the Zs have been recovered, can compute the UIJ and V IJ, therefore the expected utility But:

Non testable (no OIR) Relies on strong assumptions In particular, homoskedasticity hard to justify.

Possible solution:

consider several ‘markets’ (here cohorts), with di¤erent marginals (composition by education classes) assume ‘some’ invariance across cohorts

Underlying question: ‘did the preferences for assortative matching change’?

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Test and identi…cation (CSW 10)

Idea: structural model M =

• Z IJ, σI , µJ

holds for di¤erent cohorts c = 1, ..., T with varying class compositions. Then: sij,c = Z IJ

c + σI ˜

αJ

i,c + µJ ˜

β

I j,c

where α, β extreme value distributed, with the identifying assumption: Z IJ

c = ζI c + ξJ c + Z IJ

Interpretation: trend a¤ecting the surplus but not the supermodularity Z IJ

c Z IL c Z KJ c

+ Z KL

c

= Z IJ Z IL Z KJ + Z KL ! Null: ‘Preferences for assortativeness do not change’ Basic result: the model is (over)identi…ed

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Data

American Community Survey, a representative extract of Census. The 2008 survey has info on current marriage status, number of marriages, year of current marriage (633,885 currently married couples). Born between 1943 and 1970 for men, 1945 and 1972 Three education classes: HS drop out, HS graduate, College and above Construct 28 ’cohorts’; for each cohort, matrix of marriage proportions by classes (plus singles) Age ! assumption: husband in cohort c marries wife in cohort c + 2

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Results (1)

Estimate the Z IJs; strongly supermodular Group HSD HSG SC HSD 0.331 0.193 0.128 HSG 0.195 0.272 0.098 SC 0.028 0.233 0.468

Table: Z values: men in rows, women in columns

Variances: σ1 = .089, σ2 = .06, σ3 = .087, µ1 = .148, µ2 = .071, µ3 = .137

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Results: marital college premium

In principle, marital college premium has several components:

Marriage probability Spouse’s (distribution of) education Surplus generated Distribution of the surplus

Our estimates for women: Cohort born 1944-46 1970-72 Education HSG SC HSG SC Married 0.933 0.896 0.791 0.818 College-educated husband 0.380 0.833 0.376 0.841 Marital surplus 0.191 0.464

• 0.041

0.330 Wife’s share 0.419 0.570 0.404 0.625

Table: Marital outcomes for women in early and in recent cohorts

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Results: marital college premium

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Conclusion

1

Frictionless matching: a powerful and tractable tool for theoretical analysis, especially when not interested in frictions

2

Crucial property: intramatch allocation of surplus derived from equilibrium conditions

3

Applied theory: many applications (abortion, female education, divorce laws, children, ...)

4

Can be taken to data; structural econometric model, over identi…ed

5

Multidimensional versions: index (COQD 2010), general (GS 2010)

6

Extensions

ITU: theory; empirical applications still to be developed

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 76 / 76

Conclusion

1

Frictionless matching: a powerful and tractable tool for theoretical analysis, especially when not interested in frictions

2

Crucial property: intramatch allocation of surplus derived from equilibrium conditions

3

Applied theory: many applications (abortion, female education, divorce laws, children, ...)

4

Can be taken to data; structural econometric model, over identi…ed

5

Multidimensional versions: index (COQD 2010), general (GS 2010)

6

Extensions

ITU: theory; empirical applications still to be developed Endogenous distributions (two stage game): preferences shocks, investement in education, etc.

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 76 / 76

Conclusion

1

Frictionless matching: a powerful and tractable tool for theoretical analysis, especially when not interested in frictions

2

Crucial property: intramatch allocation of surplus derived from equilibrium conditions

3

Applied theory: many applications (abortion, female education, divorce laws, children, ...)

4

Can be taken to data; structural econometric model, over identi…ed

5

Multidimensional versions: index (COQD 2010), general (GS 2010)

6

Extensions

ITU: theory; empirical applications still to be developed Endogenous distributions (two stage game): preferences shocks, investement in education, etc. Econometrics: continuous variables

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 76 / 76

Conclusion

1

Frictionless matching: a powerful and tractable tool for theoretical analysis, especially when not interested in frictions

2

Crucial property: intramatch allocation of surplus derived from equilibrium conditions

3

Applied theory: many applications (abortion, female education, divorce laws, children, ...)

4

Can be taken to data; structural econometric model, over identi…ed

5

Multidimensional versions: index (COQD 2010), general (GS 2010)

6

Extensions

ITU: theory; empirical applications still to be developed Endogenous distributions (two stage game): preferences shocks, investement in education, etc. Econometrics: continuous variables Dynamics

P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 76 / 76